Helical biopolymers are locally rigid, but very high molecular weight materials can display flexibility approaching random coil behavior. The origin of this flexibility is not understood. In the case of double strand helical DNA, two origins for this flexibility have been proposed: 1) small fluctuations in many of the rotational angles which yield conformations with smoothly bending trajectories or 2) sharper bends whose locations depend on specific sequences. We have considered the first of these cases and have developed methods to perform averages over such conformations. Comparisons with experimentally measured persistence lengths permit determination of the average amount of torsion about backbone bonds. In the case of double strand DNA, these values are consistent with a uniformly twisting chain with mean deviations of only a 2-5 degrees. This range of fluctuations appears to be consistent with usual torsional potentials. It is expected that this flexibility would be manifested at lower molecular weights in Kerr constant measurements than for quantities such as radius of gyration.